Measuring image focusing for velocity analysis

Unbiased measure of image focusing

In Biondi (2008b), I introduced a new semblance functional, that I dubbed Image-focusing semblance, aimed at quantitatively measuring image focusing simultaneously along the spatial directions and the reflection angle (or offset) axes. The underlying idea is to extend the conventional semblance evaluation by measuring image coherency also along the the structural-dip axes. However, the estimates provided by the image-focusing semblance presented in that report can be biased by reflectors' curvature. In this section, I modify the definition of the image-focusing semblance by explicitly exposing its dependency from the image local curvature. This enables a consistent evaluation of the image focusing across both the reflection-angle axis and the structural-dip axis and improves the interpretability of the results.

The starting point of my method is an ensemble of prestack images, ${\bf R}\left({\bf x},\gamma ,\rho\right)$; these images are function of a spatial coordinate vector ${\bf x}=\left\{z,x\right\}$ (with $z$ depth and $x$ the horizontal location), the aperture angle $\gamma$, and a velocity parameter $\rho$. In the numerical examples that follow, the ensemble of prestack images is obtained by residual prestack migration in the angle domain as I presented in Biondi (2008a). The parameter $\rho$ is the ratio between the new migration velocity and the migration velocity used for the initial migration. The proposed method could be easily adapted to the case when residual prestack Stolt migration (Sava, 2003), or any other method that can efficiently generate ensembles of prestack images dependent on a velocity parameter, is used to compute ${\bf R}\left({\bf x},\gamma ,\rho\right)$. Although, when using other methods to produce the ensemble ${\bf R}\left({\bf x},\gamma ,\rho\right)$, the corrections equivalent to equations 5, 8 and 9 might be different.

To measure coherency along the structural dip $\alpha$, I first decompose the prestack image and create the dip-decomposed prestack image ${\bf R}\left({\bf x},\gamma ,\alpha ,\rho\right)$. When using either choices of residual prestack migration discussed above, the decomposition can be efficiently performed in the Fourier domain during the residual prestack migration. If other methods are used to produce the ensemble of prestack images ${\bf R}\left({\bf x},\gamma ,\rho\right)$, the dip decomposition could as efficiently performed in the space domain by applying recursive filters (Hale, 2007; Fomel, 2002). Notice, that the dip-decomposed images I use as input have different kinematic characteristics than the ones described in Reshef and Rüger (2008), Landa et al. (2008), and Reshef (2008). They obtain dip-decomposed images by not performing the implicit summation over dips that is part of angle-domain Kirchoff migration (Audebert et al., 2002), whereas I decompose the migrated images.

In equation 5 in Biondi (2008b) I defined the 2D Image-focusing semblance as:

$$S_{\left(\gamma ,\alpha \right)}\left({\bf x},\rho\right)= \frac{... ...} \sum_\gamma \sum_\alpha {\bf R}\left({\bf x},\gamma ,\alpha ,\rho\right)^2 },$$ (A-1)

where $N_{\gamma }$ and $N_{\alpha }$ are, respectively, the number of aperture angles and the number of dips to be included in the computation. The effective definition of the aperture-angle and the structural-dip ranges to be used in equation 1 is one of the practical challenges when applying the proposed method.

Measuring image focusing for velocity analysis